1. Bart M. P. Jansen Dániel Marx
Characterizing the Easy-to-Find Subgraphs from the Viewpoint of Polynomial-Time Algorithms, Kernels, and Turing Kernels
Bart M. P. Jansen Dániel Marx
January 4th, SODA 2015, San Diego

2. Does contain ? The Subgraph problem
Input: Host graph 𝐺, pattern graph 𝐻 Question: Does 𝐺 contain a subgraph isomorphic to 𝐻?
Does contain ?
For a class of graphs ℱ, the ℱ-Subgraph problem is the restriction to patterns 𝐻 from ℱ

3. Does contain 3 ? The Packing problem
Input: Host graph 𝐺, pattern graph 𝐻, integer 𝑡 Question: Does 𝐺 contain 𝑡 vertex-disjoint subgraphs, each isomorphic to 𝐻?
Does contain ?
For a class of graphs ℱ, the ℱ-Packing problem is the restriction to patterns 𝐻 from ℱ

4. Complexity depends on the type of pattern
The general Packing and Subgraph problems are NP-complete
They contain the Clique problem as a special case
However, some subcases are polynomial-time solvable
Packing vertex-disjoint 𝐾 2 ’s is the Matching problem
We investigate the question:
For which types of patterns are the Packing and Subgraph problems hard, and for which patterns are they easy?
We classify the complexity of ℱ-Packing and ℱ-Subgraph
To make this technically feasible, we focus on hereditary ℱ

5. Characterizing the complexity of a pattern
Our research aims at finding dichotomy theorems
For every hereditary family of patterns ℱ, prove that ℱ-Packing / ℱ-Subgraph is either easy or hard
Interpretation of easy and hard depends on the viewpoint
A dichotomy shows that we fully understand what makes the problem hard or easy
When proving a dichotomy, we systematically discover all the easy cases of the problem
Nontrivial easy cases appeared that were previously unknown!

6. Ingredients of our dichotomy theorems
A combinatorial characterization of the families ℱ for which the problem is easy
Algorithms for the easy cases of the problem
Hardness proofs for a small number of representative hard cases
Ramsey-theoretic argument: every family of patterns that does not satisfy the characterizing property, contains a hard subfamily

7. Polynomial-time solvability
Viewpoint I
Polynomial-time solvability

8. Polynomial-time solvable cases of ℱ-Packing
Complexity of packing connected graphs 𝐻 is well-understood
This easily extends to ℱ-Packing for hereditary families ℱ of (possibly disconnected) pattern graphs
{𝐻}-Packing is polynomial-time solvable if 𝑉 𝐻 ≤2, and NP-complete otherwise [Kirkpatrick & Hell’78]
ℱ-Packing is polynomial-time solvable if all connected components of graphs in ℱ have at most two vertices, and NP-complete otherwise

9. Polynomial-time solvable cases of ℱ-Subgraph
A graph 𝐻 is 𝑐-matching-splittable if
there is a set 𝑆 of at most 𝑐 vertices, such that
every component of 𝐻−𝑆 has at most two vertices
A graph family ℱ is matching-splittable if there is a constant 𝑐 such that every graph 𝐻∈ℱ is 𝑐-matching-splittable
ℱ-Subgraph is randomized polynomial-time solvable if ℱ is matching-splittable, and NP-complete otherwise

10. Ingredients for the ℱ-Subgraph dichotomy
Matching-splittable graph families characterize the easy cases
Randomized polynomial-time algorithm to test for the existence of a perfect matching of exactly a given weight [Mulmuley, Vazirani, Vazirani’87]
NP-completeness proofs for Triangle Packing, 𝑃 3 -Packing, Clique, and Biclique
Hereditary families ℱ that are not matching-splittable contain all cliques, bicliques, unions of triangles, or unions of 𝑃 3 ’s

11. Viewpoint II
kernelization

12. Kernelization Kernelization models provably effective preprocessing
Originated in parameterized complexity
Associates a parameter 𝑘 to every problem instance
A kernelization of size 𝑓 is a polynomial-time algorithm that reduces an input (𝑥,𝑘) to an equivalent input of size 𝑓(𝑘)
Which problems have kernels of polynomial size?

13. Kernelization for Subgraph and Packing
We choose 𝑘 to be the size of the solution
For Subgraph, 𝑘≔|𝑉 𝐻 |
For Packing, 𝑘≔𝑡⋅|𝑉 𝐻 |
For which ℱ can an instance of ℱ-Subgraph/ℱ-Packing be reduced to an equivalent one of size 𝑝𝑜𝑙𝑦(𝑘)?
“Easy ℱ”:
has a kernel whose size is a polynomial of degree depending on ℱ, but not on 𝐻
“Hard ℱ”:
no polynomial kernel whose degree is independent of 𝐻 (under some complexity-theoretic assumption)
Does contain ?

14. Kernelizable cases of ℱ-Packing
A graph 𝐻 is 𝑎-small/𝑏-thin if every connected component
has at most 𝑎 vertices, or
is a bipartite graph whose smallest side has size at most 𝑏
A graph family ℱ is small/thin if there are constants 𝑎,𝑏 such that every graph 𝐻∈ℱ is 𝑎-small/𝑏-thin
3-small/2-thin
ℱ-Packing has a polynomial kernel if ℱ is small/thin, and otherwise does not have a polynomial kernel (unless NP ⊆ coNP/poly)

15. Ingredients for the ℱ-Packing kernel dichotomy
Small/thin graph families characterize the easy cases
Sunflower lemma [Erdös & Rado‘60] for small components, novel marking procedure for thin bipartite components
Kernelization lower bounds by cross-composition and polynomial-parameter transformations from Uniform Exact Small Universe Set Cover
Hereditary ℱ that are not small/thin contain one of 8 hard families, using [Atminas, Lozin, Razgon’12]: large path implies large induced path, clique, or induced biclique

16. Kernelizable cases of ℱ-Subgraph?
We do not have a dichotomy for kernelization for ℱ-Subgraph
If such a theorem exists, it must be a lot more fragile than for case of Packing
Consider the following concrete examples:
No polynomial kernel
Polynomial kernel

17. Viewpoint III
Turing kernelization

18. Turing kernelization
A Turing kernelization of size 𝑓 is an algorithm that
decides the answer of an instance (𝑥,𝑘)
in time polynomial in 𝑥 +𝑘
when given access to an oracle that
for any instance 𝑥 ′ , 𝑘 ′ with 𝑥 ′ , 𝑘 ′ ≤𝑓 𝑘 ,
decides 𝑥 ′ , 𝑘 ′ in a single step
Reduces solving a problem to solving small instances

19. Turing kernelizable cases of ℱ-Packing
For the Packing problem, Turing kernelization is not more powerful than normal kernelization
A normal kernel also qualifies as Turing kernel
For the Subgraph problem, Turing kernelization is more powerful
ℱ-Packing has a polynomial kernel if ℱ is small/thin, and otherwise it is W[1]-hard, WK[1]-hard, or Long Path-hard

20. Turing kernelizable cases of ℱ-Subgraph
A graph 𝐻 is (𝑎,𝑏,𝑐,𝑑)-splittable if there is a set 𝑆 of at most 𝑐 vertices such that every component 𝐶 of 𝐻−𝑆:
has at most 𝑎 vertices, or
is a 𝑏-thin bipartite graph in which the closed neighborhoods of at most 𝑑 vertices are not universal to 𝑁 𝐻 𝐶 ∩𝑆
A graph family ℱ is splittable if there are constants 𝑎,𝑏,𝑐,𝑑 such that every graph 𝐻∈ℱ is (𝑎,𝑏,𝑐,𝑑)-splittable
(3,1,2,2)-splittable
ℱ-Subgraph has a polynomial Turing kernel if ℱ is splittable, and otherwise it is W[1]-hard, WK[1]-hard, or Long Path-hard

21. Conclusion
We proved dichotomy theorems for ℱ-Packing and ℱ-Subgraph
Three viewpoints: polynomial-time solvable, kernelizable, Turing kernelizable
Our work shows that proving dichotomy theorems is feasible, and aiming for them is a realistic goal
See also Dániel Marx’s MFCS talk: “Every graph is easy or hard”
Proving dichotomy theorems led us to the fact that thin bipartite graphs are easy to find and pack
Thank you!